A Bayes procedure for the identification of univariate time series models
The problem studied is the selection of a time series model from a given class of such models, using Bayesian arguments. A key result is that the posterior expected utility is maximized asymptotically by selecting the model which minimizes the criterion function \[ \Delta \{m({\hat \theta}\underset \tilde{} {\;}_ T)\}=2^{-1}Tl_ T({\hat \theta}\underset \tilde{} {\;}_ T)+2^{-1}d \ln T+\eta_ T\{m({\hat \theta}\underset \tilde{} {\;}_ T)\}, \] where T is the sample size, \({\hat \theta}_ T\) is a maximum likelihood estimator of the parameter \({\underset \tilde{} \theta}\in \Theta\) (so that asymptotically the likelihood function \(l_ T\) behaves like the kernel of a d variate Gaussian density with mean vector \({\hat \theta}_ T)\), \(m({\underset \tilde{} \theta})\) denotes the action of choosing \({\underset \tilde{} \theta}\in \Theta\), and \(\eta_ T\) is the integrated square relative error, \(\eta_ T=\int^{\pi}_{- \pi}[f(\omega)/g({\underset \tilde{} \theta},\omega)-1]^ 2 d\omega\), where \(f(\omega)\) is the power spectrum and \(g({\underset \tilde{} \theta},\omega)\) a particular functional parametric form. In a corollary, \(\Delta\) is specified for choosing one ARMA(p,d) model, and the general and particular criteria are related to the work of other researchers. In terms of ARMA models the author expresses that ''the criterion function leads to the selection of the model that appears to provide the best compromise between predictive characteristics, the autocorrelation structure of the residuals and model dimensionality''. In a final section consistency of the criterion is proved.
one-step ahead prediction error variance
autocorrelations of the residual process
scientific article; zbMATH DE number 3974122